#8258630

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,6,7)(3,8,4,5)(9,10)(11,13)(12,15)(14,16)(17,18,22,23)(19,25,27,20)(21,24,29,30)(26,31,32,28);(2,7)(3,4)(10,11)(14,15)(18,23)(20,25)(24,30)(28,31);(2,7)(5,8)(9,10,13,11)(12,14,16,15)(21,29)(24,30)(26,32)(28,31);(1,3,6,4)(2,5,7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31);(1,4,6,3)(2,8,7,5)(9,12)(10,14)(11,15)(13,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,31)(30,32);(1,5,6,8)(2,3,7,4)(17,21,22,29)(18,24,23,30)(19,26,27,32)(20,28,25,31)

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8165826);